20061ee131A_1_131ahw8

# 20061ee131A_1_131ahw8 - with a mean of 50 If the variance...

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EE 131A Homework #8 Winter 2006 Due March 16th K. Yao Read pp. 69-73; 191-246 1. A miner is trapped in a mine containing 3 doors. The ﬁrst door leads to a tunnel that will take him to safety after 3 hours of travel. The second door leads to a tunnel that will return him to the mine after 5 hours of of travel. The third door leads to a tunnel that will return him to the mine after 7 hours of travel. If we assume the miner is all times equally likely to chose any one of the door, what is the expected length of time he reaches safety? Hint: Let X denote the amount of time (in hours) until the miner reaches safety and let Y denote the door he initially chooses. Find E { X | Y = y j } and then use E { X } = J j =1 E { X | Y = y j } . 2. The r.v. X and Y have a joint pdf given by f X,Y ( x,y ) = 2 e 2 x /x, 0 x < , 0 y x ; f X,Y ( x,y ) = 0 , otherwise . Compute cov ( X,Y ) . Hint: R 0 y n - 1 e - y dy = Γ( n ) = ( n - 1)! . 3. Suppose it is known that the number of items produced in a factory during a week is a r.v.
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Unformatted text preview: with a mean of 50. If the variance of the r.v. is known to be 25, then what is a lower bound on the probability that this week’s production will be between 40 and 60? 4. Problem 81, page 182. 5. A voltage V is a function of time t and is given by V ( t ) = X cos(2 πft )+ Y sin(2 πft ) , in which f is the frequency (in Hz) and X and Y are two independent Gaussian zero mean r.v.’s of variance σ 2 . a. Show V ( t ) can be written as V ( t ) = R cos(2 πft-Θ) . Hint: deﬁne R = ( X 2 + Y 2 ) . 5 . b Show R is a Rayleigh r.v. and Θ is uniformly distributed in [0 , 2 π ) , and these two r.v.’s are independent. 6. Problem 22, page 320. Hint: Use CLT and Gaussain r.v. to approximate the Binomial r.v....
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